Proposition VII.20 — The least numbers of those which have the same ratio with th…

Proof

Given

a : b = p : q

with

p

q

least in their ratio, by VII.2

g cd (a, b) \cdot p = a

and

g cd (a, b) \cdot q = b

, so

p

q

each measure

a

b

the same number of times

g cd (a, b)

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Depends on (2)

VII.2Proposition VII.2Given two numbers not prime to one another, to find their greatest common measure.
VII.19Proposition VII.19If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced…

Required by (dependents) (7)

VII.21Proposition VII.21Numbers prime to one another are the least of those which have the same ratio with them.
VII.22Proposition VII.22The least numbers of those which have the same ratio with them are prime to one another.
VII.34Proposition VII.34Given two numbers, to find the least number which they measure.
VIII.1Proposition VIII.1If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the…
VIII.3Proposition VIII.3If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the…
VIII.6Proposition VIII.6If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will…
VIII.7Proposition VIII.7If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the…

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